Spectral Analysis in Rigged Hilbert Spaces

Abstract

In Quantum Physics, in particular in the framework of Dirac’s bra and ket formulation, the existence and completeness of “generalized eigenfunctions” plays a prominent rôle. This chapter explains the most important mathematical aspects on which this formalism is based, the so called “nuclear spectral theorem.” Naturally some preparations are needed. We begin by explaining the concept of a “rigged Hilbert space” also called a “Gelfand triple”. This requires some background on “nuclear spaces”. For the subsequent analysis the most important result is the structure of the natural embedding in a Gelfand triple. In order to be able to state the classical spectral representations in the generality which we need, the concept of a “direct integral of Hilbert spaces” has to be introduced. This then allows to state the “functional form” of the classical spectral representation of self-adjoint operators in a separable Hilbert space which is combined with the result on the structure of the natural embedding in a Gelfand triple and thus leads to the result on the existence and completeness of generalized eigenfunctions.